新Mathieu-Duffing 系統的渾沌現象與其應用適應逆步控制及部分區域穩定理論之實用混合投影渾沌同步及辛渾沌同步

國 立 交 通 大 學

機械工程學系

碩士論文

新 Mathieu-Duffing 系統的渾沌現象與其應用

適應逆步控制及部分區域穩定理論之

實用混合投影渾沌同步及辛渾沌同步

Chaos and Pragmatical Hybrid Projective

and Symplectic Chaos Synchronization of

a New Mathieu-Duffing System by Adaptive

Backstepping Control and by Partial Region

Stability Theory

研 究 生：李彥賢

指導教授：戈正銘 教授

中華民國九十七年六月

新 Mathieu-Duffing 系統的渾沌現象

與其應用適應逆步控制及部分區域穩定理論之

實用混合投影渾沌同步及辛渾沌同步

Chaos and Pragmatical Hybrid Projective and Symplectic Chaos

Synchronization of a New Mathieu-Duffing System by Adaptive

Backstepping Control and by Partial Region Stability Theory

研究生：李彥賢 Student: Yan-Sian Li

指導教授：戈正銘 Advisor: Zheng-Ming Ge

國 立 交 通 大 學

機 械 工 程 研 究 所

碩 士 論 文

A Thesis

Submitted to Institute of Mechanical Engineering College of Engineering

National Chiao Tung University In Partial Fulfillment of the Requirement

For the Degree of master of science In

Mechanical Engineering June 2008

Hsinchu, Taiwan, Republic of China

新 Mathieu-Duffing 系統的渾沌現象與其應用

適應逆步控制及部分區域穩定理論之

實用混合投影渾沌同步及辛渾沌同步

學生:李彥賢 指導教授：戈正銘

摘要

本論文由三部分構成：（1）以相圖、龐卡萊映射圖、分岐圖、功率譜及 Lyapunov 指數圖等數值方法研究 Mathieu – Duffing 系統的渾沌行為。（2）用適應逆步控制在 不同的初始條件下的兩個 Mathieu – Duffing 雙系統對不同的渾沌系統 Duffing – van der Pol 系統與 系統的實用混合投影渾沌同步及辛渾沌同步。（3）應用部分區域 穩定性理論研究廣義渾沌同步、渾沌控制及實用混合投影渾沌同步。

Chaos and Pragmatical Hybrid Projective

and Symplectic Chaos Synchronization of

a New Mathieu-Duffing System by Adaptive

Backstepping Control and by Partial Region

Stability Theory

Student：Yan-Sian Li Advisor：Zheng-Ming Ge

ABSTRACT

This thesis consists of three parts: (1) the chaotic behaviors of are studied numerically by phase portraits, Poincaré maps, bifurcation diagrams, power spectrum and Lyapunov exponent diagrams. (2)

system is studied for pragmatical hybrid projective hyperchaotic generalized synchronization (PHPHGS) and pragmatical hybrid projective and symplectic synchronization (PHPSS) with different kinds of different chaotic systems, Duffing-van der Pol system and system, by adaptive backstepping control. (3) chaotic generalized synchronization, chaos control and pragmatical hybrid projective generalized synchronization is studied by partial region stability theory.

Mathieu - Duffing system

Mathieu - Duffing

誌謝

本論文得以完成，首先感謝我的指導老師戈正銘教授兩年來悉心的指 導。老師溫文儒雅的學者風範，兼具科學家的敏銳及文人的氣息，都是現代 理工背景出身的人所罕見的。嚴謹的治學態度興追求新知的熱誠，將是我永 續學習的榜樣。 在兩年的碩士生涯裡，感謝博士班楊振雄、張晉銘、李仕宇學長，碩士 班李乾豪、吳宗訓、李式中、林森生學長及翁郁婷學姊，在我研究遇到瓶頸 時，給予我寶貴的意見及分享個人人生經驗；也感謝我的同學何俊諺、許凱 銘、陳聰文…等，大家彼此互相扶持成長，共同度過這兩年研究的時光，留 下許多快樂的回憶。另外要感謝學弟陳志銘、徐瑜韓、張育銘，幫忙處理繁 瑣雜事，得以讓我們專心致力於研究。 最後，感謝我的父親李明崇先生、母親李惠貞女士任勞任怨地為我付出， 有您們的全力支持讓我攻讀碩士學位，使我無後顧之憂地專致於研究上，雖 然您們都不在我身邊，但我知道您們是非常關心我的，常常透過電話表達對 我的關心，都讓我感到非常的溫暖，給我勇氣迎接挑戰。也感謝我親愛的哥 哥李彥輝、賢慧的大嫂張靜怡及健壯的弟弟李彥霆，一直以來都給我許多寶 貴的意見與開導，我愛我的家人，因為有您們，讓我感覺到有家的溫暖，也 感謝您們的教養，得以讓我順利拿到碩士學位。在此也謝謝女友邱佳儀小姐 對我的關心、體諒及支持。最後，僅以此論文獻給你們大家。

CONTENTS

Abstract………i Acknowledgement……….iii Contents……….iv List of figures……….………....vi Chapter 1 Introduction………..1

Chapter 2 Chaos in a New Mathieu – Duffing System………...………4

2.1 Preliminaries………..4

2.2 A new Mathieu – Duffing system………..4

2.3 Simulation results………...5

Chapter 3 Pragmatical Hybrid Projective Hyperchaotic Generalized of Hyperchaotic Systems by Adaptive Backstepping Control………9

3.1 Preliminaries………..9

3.2 Synchronization scheme………9

3.3 Numerical results of PHPHGS by adaptive backteppting control…………..11

Chapter 4 Pragmatical Hybrid Projective and Symplectic Synchronization of Different Order Systems with New Control Lyapunov Function by Adaptive Backstepping Control………..22

4.1 Preliminaries………22

4.2. Symplectic synchronization scheme………..23

4.3. Lü system……….25

4.4. Numerical results of CHPS by adaptive backstepping control………..26

Chapter 5 Chaos Control of a New Mathieu- Duffing System by GYC Partial Region Stability Theory………...36

5.2 Chaos control scheme………..36

5.3 Numerical simulations of chaos control by GYC ……….37

Chapter 6 Chaos Generalized Synchronization of New Mathieu- Duffing Systems and Chaotization by GYC Partial Region Stability Theory ………47

6.1 Preliminaries………47

6.2 Chaos generalized synchronization strategy……….47

6.3 Numerical simulations of chaos generalized synchronization by GYC theory………..………48

6.4 Chaotization to a new Mathieu –Duffing system………..55

Chapter 7 Chaos of a New Mathieu – Duffing System with Bessel Function Parameters ………...67

7.1 Preliminaries………67

7.2 A new Mathieu – Duffing system with Bessel function parameters…….…...67

7.3 Simulation results………....69

Chapter 8 Pragmatical Hybrid Projective Generalized Synchronization of New Mathieu- Duffing Systems with Bessel Function Parameters by Adaptive Control and GYC Partial Region Stability Theory………....76

8.1 Preliminaries………76

8.2 Synchronization scheme………..76

8.3. Numerical results of PHPGS by GYC partial region stability theory……...78

Chapter 9 Conclusions……….85

Appendix A GYC Pragmatical Asymptotical Theorem………88

Appendix B GYC Partial Region Stability Theory………...91

LIST OF FIGURES

Fig. 2.1 The time history of the four states. ………6

Fig. 2.2 The phase portraits and Poincaré maps of x x dimensions……….6 ₁, ₂ Fig. 2.3 The phase portraits and Poincaré maps of x x dimensions………7 ₃, ₄ Fig. 2.4 Chaotic phase protrait of a new Mathieu-Duffing system in three dimensions….7 Fig. 2.5 The chaotic power spectrum of x for a new Mathieu-Duffing system………...8 ₁ Fig. 2.6 Lyapunov exponents of a chaotic new Mathieu-Duffing system (+,0,0,-)……….8

Fig. 3.1 Chaotic phase portrait of Duffing – Van der Pol system………..18

Fig. 3.2 The time histories of errors (e₁, e₂, e₃, e₄).……….18

Fig. 3.3 The time histories of estimated parameter aˆ . ………..19

Fig. 3.4 The time histories of estimated parameter bˆ.……….19

Fig. 3.5 The time histories of estimated parameter cˆ ……….20

Fig. 3.6 The time histories of estimated parameter dˆ.……….20

Fig. 3.7 The time histories of estimated parameter ˆe.………..……21

Fig. 3.8 The time histories of estimated parameter ˆf.……….21

Fig. 4.1 Chaotic phase protract of Lu system……….33

Fig. 4.2 Synchronization error for traditional control Lyapunov function from 600s ~ 1000s……….33

( ) T V e =e e Fig. 4.3 Synchronization error for new control Lyapunov function from 600s ~ 1000s.………34

( ) exp( T ) 1 V e = ke e − Fig. 4.4 The time histories of estimated parameter aˆ and ……….…34 bˆ Fig. 4.5 The time histories of estimated parameter cˆ and dˆ.………35

Fig. 4.6 The time histories of estimated parameter ˆe and ˆf.………35

Fig.5.2 Time histories of x x x x for Case I. ………...43 ₁, ₂, ₃, ₄

Fig. 5.3 Phase portrait of error dynamics for Case II. ………...44

Fig. 5.4 Time histories of errors for Case II. ……….44

Fig. 5.5 Time histories of x x x x for Case II………....45 ₁, ₂, ₃, ₄ Fig. 5.6 Phase portrait of error dynamics for Case III………...45

Fig. 5.7 Time histories of errors for Case III……….46

Fig. 5.8 Time histories of x x x x z z z z_{1, 2, 3, 4, 1, 2, 3, 4} for Case III. ……….……...46

Fig. 6.1 Phase portrait of four errors dynamics for Case I. ………...59

Fig.6.2 Time histories of errors for Case I. ………59

Fig. 6.3 Time histories of x x x x y y y y_{1, 2, 3, 4, 1, 2, 3, 4}for Case I. ………60

Fig. 6.4 Phase portrait of error dynamics for Case II. ………...60

Fig. 6.5 Time histories of errors for Case II. ……….61

Fig. 6.6 Time histories of x_i− + and y_i k_i −msinwt for Case II. ……….61

Fig. 6.7 Phase portrait of error dynamics for Case III. ………..62

Fig. 6.8 Time histories of errors for Case III. ………62

Fig. 6.9 Time histories of ₁ 2 2xi and y for Case III. ……….63 i Fig. 6.10 Phase portrait of error dymanics for Case IV. ………63

Fig. 6.11 Time histories of errors for Case IV. ………...64

Fig. 6.12 Time histories of x_i− + and y_i k_i − for Case IV. ………64 z_i Fig. 6.13 Phase portraits of a new periodic system. ………...65

Fig. 6.14 Phase portraits of error dynamics for Section 6.4. ………65

Fig. 6.15 Time histories of errors. ………66

Fig. 7.1 The time history of a(t) with 50 sec.………70

Fig. 7.2 The time history of b(t) with 50 sec.………70

Fig. 7.3 The time history of c(t) with 50 sec.……….71

Fig. 7.4 The time history of d(t) with 50 sec.………71

Fig. 7.5 The time history of e(t) with 50 sec.……….72

Fig. 7.6 The time history of f(t) with 50 sec.……….72

Fig. 7.7 The time history of the four states with 5000 sec……….73

Fig. 7.8 The phase portrait and Poincaré mapof x x dimensions………..73 ₁, ₂ Fig. 7.9 The phase portrait and Poincaré map of x x dimensions……….74 ₃, ₄ Fig. 7.10 The chaotic power spectrum of x for Mathieu-Duffing system. …………74 ₁ Fig. 7.11 Bifurcation of chaotic Mathieu-Duffing system. ………...75

Fig. 7.12 Lyapunov exponents of chaotic Mathieu-Duffing system (+,0,-,-). …………..75

Fig. 8.1 Phase portraits of error dynamics. ………...82

Fig. 8.2 The time histories of errors (e₁, e₂, e₃, e₄).……….82

Fig. 8.3 The time histories of a and b .……….83 

Fig. 8.4 The time histories of c and d.……….83

Fig. 8.5 The time histories of e and f.……….84 Fig. B.1 Partial regions Ω and Ω ………..99 ₁

Chapter 1

Introduction

Chaotic system features that it has complex dynamical behaviors and sensitive behavior dependence initial conditions. Since Pecora and Carroll [1] introduced a method to synchronize two identical systems with different initial conditions, chaos synchronization has attracted a great deal of attention from various fields during the last two decades. Recently, many valuable control methods and techniques have been developed to synchronize chaotic systems, such as backstepping design method [2], impulsive control method [3], invariant manifold method [4], adaptive control method [5], linear and nonlinear feedback control method [6], and active control approach [7], PC method [8], etc. Most of them are based on the exact knowledge of the system structure and all parameters. But in practice, some or all of the system parameters are uncertain. Additionally, these parameters change at every time. A lot of researchers have studied to solve this problem by adaptive synchronization [12-15].

This thesis is organized as follows. In Chapter 2, the chaotic behaviors of a new Mathieu – Duffing system is studied numerically by phase portraits, Poincaré maps and

Lyapunov exponent diagrams. That can be explained that chaos exists in the new

Mathieu- Duffing system.

In the current scheme of adaptive synchronization [12-15], the traditional Lyapunov stability theorem and Babalat lemma are used to prove that the error vector approaches zero, as time approaches infinity. But the question of that why the estimated parameters also approach uncertain parameters remains unanswered. In Chapter 3, by the pragmatical asymptotical stability theorem, the question can be answered strictly. That the error vector tends to zero and the estimated parameters approach uncertain values is guaranteed by the pragmatical asymptotical stability theorem [17, 18] and adaptive

backstepping control.

In Chapter 4, a new kind of synchronization and a new control Lyapunov function are proposed. The symplectic synchronization

( , , , )

y=H x y z t (1.1)

is studied, where x and y are the state vectors of the ‘‘master’’ and of the ‘‘slave’’ respectively and z is a given function vector of time, which may take various forms, either a regular or a chaotic function of time [22-24]..

A new control Lyapunov function ( ) exp( T ) 1

V e = ke e − (1.2)

is proposed for backstepping control where e is error dynamics. Using the new control Lyapunov function, the error tolerance can be decreased marvelously to ₁₀−17 _{of that} using traditional control Lyapnov function

( ) T

V e =e e (1.3)

In Chapter 5, a new scheme to achieve chaos control by partial region stability theory is proposed [33, 34]. By using the GYC partial region stability theory, Lyapunov function becomes a simple linear homogeneous function of error states and controllers are simpler and introduce less simulation error. Similarly, in Chapter 6, a new chaos generalized synchronization strategy by partial region stability theory is proposed [33, 34].

In Chapter 7, the chaotic behaviors of a new Mathieu – Duffing systems with Bessel function parameters is studied numerically by phase portraits, Poincaré maps,bifurcation

and Lyapunov exponent diagrams. If can be discovered that chaos also exists in the new

Mathieu – Duffing systems with Bessel function parameters.

In Chapter 8, the error vector tends to zero and that the estimated parameters approach uncertain values is guaranteed by the pragmatical asymptotical stability

theorem [17, 18] and GYC partial region stability theory. A pragmatical hybrid projective generalized synchronization (PHPGS) strategy is proposed.

Chapter 2

Chaos in a New Mathieu – Duffing

System

2.1 Preliminaries

Abundant chaotic behaviors in a new Mathieu - Duffing system are studied numerically by phase portraits, Poincaré maps and Lyapunov exponent diagram.

2.2 A new Mathieu – Duffing system

Mathieu system and Duffing system are two typical nonlinear nonautonomous systems:

1 2

3

2 ( sin ) 1 ( sin ) 1 2 sin

d x x dt d x a b t x a b t x cx d t dt ω ω ω ⎧ ₌ ⎪⎪ ⎨ ⎪ _{= − + − + − +} ⎪⎩

(2.1) 3 4 3 4 3 3 4 sin d x x dt d x x x ex f t dt ω ⎧ ₌ ⎪⎪ ⎨ ⎪ _{= − − − +} ⎪⎩

(2.2)

Exchanging sin tω term in Eq.(2.1) with x and ₃ sin tω in Eq.(2.2) with x₁, we

1 2 3 2 3 1 3 1 2 3 3 4 3 4 3 3 4 1 ( ) ( ) d x x dt d x a bx x a bx x cx dx dt d x x dt d x x x ex fx dt ⎧ ₌ ⎪ ⎪ ⎪ _{= − + − + − +} ⎪⎪ ⎨ ⎪ ₌ ⎪ ⎪ ⎪ = − − − + ⎪⎩

(2.3)

wherex₁,x₂,x ,₃ x₄ are state variables, and a b c d e f are parameters. We analyses , , , , ,

and presents simulation results of the chaotic dynamics produced from a new Mathieu – Duffing system in the state equations of Eq. (2.3). When a=20.30, b=0.5970, c=0.005,

d =-24.441, e=0.002, f =14.63, abundant chaotic behaviors are shown by phase

portraits, Poincaré maps, power spectrum and Lyapunov exponent diagram in Figs 2.1-2.6.

2.3 Simulation results

The parameters used are: a=20.30, b=0.5970, c=0.005, d=-24.441, e=0.002,

and f =14.63. The numerical simulations are carried out by MATLAB with using the fractional operator in the Simulink environment.

The time history of four states, phase portraits, Poincaré maps and power spectrum

of a new Mathieu – Duffing system with parameters given are shown in Fig.2.1~Fig.2.5.

Chaos exists for all cases.

We vary the system parameter d, with other system fixed as: a=20.30, b=0.5970, c=0.005, e=0.002, f =14.63, the chaotic Lyapunov exponents are obtained as shown

Fig. 2.1 The time history of the four states.

Fig. 2.3 The phase portraits and Poincaré maps of x x dimensions.₃, ₄

Fig. 2.5 The chaotic power spectrum of x for a new Mathieu-Duffing system. ₁

Chapter 3

Pragmatical Hybrid Projective Hyperchaotic

Generalized Synchronization of Hyperchaotic Systems

by Adaptive Backstepping Control

3.1Preliminaries

In the current scheme of adaptive synchronization [11-15], the traditional Lyapunov stability theorem and Babalat lemma are used to prove that the error vector approaches zero, as time approaches infinity. But the question of that why the estimated parameters also approach uncertain parameters remains unanswered. By the pragmatical asymptotical stability theorem, the question can be answered strictly. In this Chapter, that the error vector tends to zero and the estimated parameters approach uncertain values is guaranteed by the pragmatical asymptotical stability theorem [16, 17] and adaptive backstepping control.

3.2 Synchronization scheme

Among many kinds of synchronizations [2-6], the generalized synchronization is investigated [7-8]. This means that we can give a functional relationship between the states of the master and slavey=G x( ). In this Chapter, a special case of hybrid projective hyperchaotic generalized synchronization

( ) ( ) ( )

y=G x =g x t z t (3.1)

is studied, where x and y are state variable vectors of the master and slave,

respectively. z is state vector of a third hyperchaotic system, called functional system, since it is a constituent of function G.. While the entries of constant vector g can be either positive or negative, hybrid projective synchronization is named. When z is chaotic, chaotic generalized synchronization is named. Pragmatical asymptotical stability theorem is used, pragmatical synchronization is named. As a whole,pragmatical hybrid projective hyper-chaotic generalized synchronization (PHPHGS) is named.

The master system is ( )

x= f x (3.2)

where [ , ,..., ]_{1 2} T n n

x= x x x ∈ℜ is a state vector and all parameters of Eq.(3.2) are

uncertain. The slave system is ( )

y= f y + (3.3) u

1 2

[ , ,...., ]T n n

y= y y y ∈ℜ is also a state vector, where u is a control vector. The functional

system is ( )

z=k z (3.4)

where [ , ,..., ]_{1 2} T n n

z= z z z ∈ℜ is a hyperchaotic state vector and all parameters of

Eq.(3.4) are known.

The control task is to force the slave state vector to track an n-dimensional desired vector

1 2 1 1 1 2 2 2

( ) [ ( ), ( ),..., ( )] [_n ( ) ( ), ( ) ( ),..., _{n n}( ) ( )]_n

h t = h t h t h t = g x t z t g x t z t g x t z t (3.5)

where g=[ , ,...,g g_{1 2} gn] are constant vector with positive and negative entries. Define the error vector ( ) [ , ,..., ]_{1 2} T n

n

e t = e e e ∈ℜ :

( ) ( ) ( )

The controlling goal is that 0 lim = ∞ → e t (3.7) can be accomplished on the base of pragmatical asymptotical stability theorem by adaptive backstepping control.

3.3. Numerical results of PHPHGS by adaptive backstepping

control

Take the Mathieu-Duffing system (2.3) as master system, where a, b, c, d, ,

e f are uncertain parameters and following Mathieu-Duffing system as slave system:

1 2 3 2 3 1 3 1 2 3 3 4 3 4 3 3 4 1 ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ d y y dt d y a by y a by y cy dy dt d y y dt d y y y ey fy dt ⎧ ₌ ⎪ ⎪ ⎪ _{= − + − + − +} ⎪ ⎨ ⎪ ₌ ⎪ ⎪ ⎪ = − − − + ⎩ (3.8)

where a b c d e and fˆ, , , ,ˆ ˆ ˆ ˆ ˆare estimated parameters. Finally, take a new Duffing – van der Pol system [32]: 1 2 3 2 1 1 1 2 1 3 3 4 2 4 1 3 1(1 3) 4 1 1 d z z dt d z z z a z d z dt d z z dt d z b z c z z f z dt ⎧ ₌ ⎪ ⎪ ⎪ _{= − − − +} ⎪⎪ ⎨ ⎪ ₌ ⎪ ⎪ ⎪ = − + − + ⎪⎩ (3.9)

constants. When a₁=0.01, b₁=1.00063, c₁=5, d₁=0.66635, f₁=0.05, chaos occurs as

shown in Fing 3.1.

By definition, error states (e ii =1, 2,3, 4) are 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 e y g x z e y g x z e y g x z e y g x z = − ⎧ ⎪ = − ⎪ ⎨ = − ⎪ ⎪ = − ⎩ (3.10)

where g g g and g are partly positive and partly negative constants, to form hybrid ₁, , ,_{2 3 4} projective synchronization.

In order to lead( , , , )y y y y_{1 2 3 4} to(g x z g x z g x z g x z_{1 1 1}, _{2 2 2}, _{3 3 3}, _{4 4 4}) , u u u and ₁, ,_{2 3} u₄

are added to each equation of Eq.(3.8), respectively:

1 2 1 3 2 3 1 3 1 2 3 2 3 4 3 3 4 3 3 4 1 4 ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ d y y u dt d y a by y a by y cy dy u dt d y y u dt d y y y ey fy u dt ⎧ _{= +} ⎪ ⎪ ⎪ _{= − + − + − + +} ⎪ ⎨ ⎪ _{= +} ⎪ ⎪ ⎪ = − − − + + ⎩ (3.11)

The error dynamics becomes:

1 2 2 2 2 1 2 1 1 1 2 1 3 2 1 1 1 1 1 3 3 3 3 1 1 1 1 3 1 1 1 3 3 3 1 2 2 2 2 3 3 3 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3 3 3 3 1 1 1 3 3 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 3 3 3 ˆ ˆ e e g x z g x z g x z u e ae ag x z be e bg x z e bg x z e bg x z g x z ae ag x z e ag x z e ag x z be e bg x z e e bg x z e e bg x z e bg x z = + − − + = − − − − − − − − − − − − − − − 3 2 2 2 2 3 1 1 1 1 3 3 3 1 1 1 1 3 3 3 1 3 3 3 1 1 1 3 3 3 2 2 2 2 3 3 3 3 2 1 2 2 3 1 2 3 3 3 2 1 2 2 3 1 2 2 2 2 2 3 2 2 2 1 2 2 1 1 2 2 2 1 2 2 3 2 3 4 4 4 4 3 4 3 ˆ ˆ 3 3 ˆ _{ˆ ˆ} ˆ ˆ e bg x z g x z e bg x z g x z e bg x z g x z ce cg x z de dg x z ag x z bg x x z ag x z bg x x z cg x z dg x z g x z g x z a g x z d g x z u e e g x z g x z − − − − − + + + + + + + − + + + − + = + − _{3 3 4 3} 3 2 2 2 2 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 3 1 1 1 1 4 3 4 4 3 4 4 4 4 4 1 4 1 4 4 3 2 1 4 4 4 1 4 4 3 4 1 4 4 1 4 ˆ ˆ 3 3 ˆ ˆ g x z u e e g x z e e g x z e g x z g x z ee eg x z fe fg x z g x z g x z eg x z fg x z b g x z c g x z c g x z z f g x z u ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ − + ⎪ = − + − − − − − − ⎪ ⎪ _{+ + + + + − +} ⎪ ⎪ _{− + − +} ⎪⎩ (3.12)

where

i i i i i i i i

e = −y g x z −g x z , (i=1, 2,3, 4) (3.12)

Step1. We consider the asymptotical stability of e₁=0:

1 2 2 2 2 1 2 1 1 1 2 1

e = +e g x z −g x z −g x z +u (3.13)

where

e

₂ is regarded as a controller. Choose a control Lyapunov function (CLF) of the form

2 1 1 2 1

v = e (3.14)

Its time derivative along the solution of Eq.(3.13) is 1 1( 2 2 2 2 1 2 1 1 1 2 1)

v =e e +g x z −g x z −g x z +u (3.15)

Assume virtual controllere₂ =α₁( )e₁ = −e₁. Eq.(3.15) can be rewritten as

1 1( 1 2 2 2 1 2 1 1 1 2 1) v =e − +e g x z −g x z −g x z +u (3.16) Choose 1 2 2 2 1 2 1 1 1 2 u = −g x z +g x z +g x z (3.17) Eq.(3.16) becomes 2 1 1 0 v = − < (3.18) e

This means that e₁=0 is asymptotically stable.

Step2. When

e

₂ is considered as a controller,

α

₁

(

e

₁

)

is an estimative function.

Define

2 2 1( )1 1 2

w = −e α e = +e e (3.19)

Study the ( ,e w_{2 2}) system. We have

2 1 2

w = +e e (3.20)

2 2 2 2 2 1

2 1 2( 2 ) 0

v = +v w +a +b +c +d > , (3.21)

where a= −a aˆ,b= −b bˆ,c= −c cˆ,d = −d dˆ and ˆa,ˆb, ˆc, ˆd are estimated values of

the unknown parameters a, b, c, d, respectively. We have

2 1 2 1 2 3 1 2 2 1 1 1 1 3 3 3 1 1 1 1 3 3 3 1 2 2 2 2 3 3 3 3 2 1 1 1 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 3 3 3 1 2 2 2 3 3 1 1 1 3 3 3 1 1 1 ( ) ˆ ˆ ˆ ˆ ˆ ( ˆ ˆ ˆ ˆ ˆ 3 3 3 ˆ ˆ 3 v v w e e aa bb cc dd v w e ae ag x z bg x z e bg x z g x z ae ag x z e ag x z e ag x z bg x z e bg x z g x z e bg x z g x z e bg x = + + + + + + = + − − − − − − − − − − − − 3 1 3 3 3 2 2 2 2 3 3 3 2 1 2 3 3 3 2 3 1 2 2 1 2 2 3 1 2 2 2 2 2 3 2 2 2 1 2 2 1 1 2 2 2 1 2 2 3 2 3 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 ˆ ˆ ˆ ) ˆ ˆ ˆ ˆ ˆ ( 3 3 z g x z ce cg x z dg x z ag x z bg x x z ag x z bg x x z cg x z dg x z g x z g x z a g x z d g x z u aa bb cc dd w e be bg x z be bg x z e bg x z e − − + + + + + + − + + + − + + + + + + − − − − − 3 3 3 1−bg x zˆ 1 1 1 +dˆ) (3.22)

Choose controllere3 =α2( ) 0e1 = , and choose 3 2 1 2 2 2 1 2 2 3 2 1 3 2 2 2 1 3 2 2 2 2 2 2 2 3 2 2 3 2 2 2 1 1 1 1 3 3 3 1 1 1 1 3 3 3 1 1 1 1 1 2 2 2 3 3 3 3 1 1 1 1 1 1 1 3 3 3 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3ˆ ˆ ˆ ˆ 3 a a g x z w g x z w b b g x x z w g x x z w c c g x z w d d g x z w u e ae ag x z bg x z e bg x z g x z ae ag x z e ag x z e ag x z bg x z e = − = − − = − = − − = − = − = − = = − + + + + + + + + + + 2 1 1 1 3 3 3 1 2 2 2 3 3 3 1 1 1 3 3 3 1 1 1 1 3 3 3 2 2 2 2 3 3 3 2 1 2 3 3 3 2 3 1 2 2 1 2 2 3 1 2 2 2 2 2 3 2 2 2 1 2 2 1 1 2 2 2 1 2 2 3 2 ˆ 3 ˆ ˆ _{ˆ ˆ} ˆ _ˆ 3 ˆ _ˆ ˆ _ˆ ˆ bg x z g x z e bg x z g x z e bg x z g x z ce cg x z dg x z ag x z bg x x z ag x z bg x x z cg x z dg x z g x z g x z a g x z d g x z w ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ _{+ + + + − −} ⎪ ⎪ _{− − − − + − −} ⎪ − + − ⎪ ⎪⎩ (3.23) Eq.(3.22) becomes 2 2 1 2 0 v = −v w < (3.24)

This means that e₂ =0 is asymptotically stable.

Step3. When e is considered as a controller, ₃ α₂( )e₁ is estimative function.

Define

3 3 2( )1 3

w = −e α e = (3.25) e

3 3

w = (3.26) e

Choose a control Lyapunov function of the form 2

1

3 2 2 3 0

v = +v w > (3.27)

Its time derivative through the third equation of Eq.(3.11) is 3 2 3( 4 4 4 4 3 4 3 3 3 4 3)

v = +v w e +g x z −g x z −g x z +u (3.28)

where e₄ is a virtual controller. Take

4 3( )1 3 3 e =α e = −w = − (3.29) e and choose 3 4 4 4 3 4 3 3 3 4 u = −g x z +g x z +g x z (3.30) Eq.(3.28) becomes 2 3 3 3 0 v = −v w ≤ (3.31)

This means that e₃= is asymptotically stable. 0

Step4. When e₄ is a virtual controller, α₃( )e₁ is estimative function.

Define 4 3 3( )1 3 4 w = −e α e = + (3.32) e e then 4 3 4 w = + (3.33) e e

Choose a Lyapunov function of the form 2 2 2

1

4 3 2( 4 ) 0

V =v = +v w +e + f > (3.34)

Its time derivative is

3 2 2 2 2 3 3 3 4 3 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 4 4 4 1 1 1 1 4 3 4 4 3 4 4 4 4 4 1 4 2 1 4 4 3 1 4 4 4 1 4 4 3 4 1 4 4 1 4 ˆ ( 3 3 ˆ ˆ ˆ ) V v v w e e g x z e e g x z e g x z g x z ee eg x z fe fg x z g x z g x z eg x z fg x z b g x z c g x z c g x z z f g x z u ee ff = = + − − − − − − − − + + + + + − + − + − + + + (3.35) Choose

4 4 4 4 4 1 4 4 3 2 2 2 2 3 3 3 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 4 4 4 1 1 1 1 4 3 4 4 3 4 4 4 4 4 1 4 2 1 4 4 3 1 4 4 4 1 4 4 3 4 1 4 4 1 4 ˆ ˆ ˆ 3 3 ˆ ˆ ˆ ˆ ˆ e e g x z w f f g x z w u e e g x z e e g x z e g x z g x z ee eg x z fe fg x z g x z g x z eg x z fg x z b g x z c g x z c g x z z f g x z w ⎧ = − = − ⎪ ⎪ = − = ⎪⎪ _{= − + + + + + + +} ⎨ ⎪ + − − − − − + − + − + − ⎩ ⎪ ⎪ ⎪ (3.36) Eq.(3.35) becomes 2 4 3 4 2 2 2 2 1 2 3 4 0 V v v w e w w w = = − = − − − − < (3.37)

This means that e4 =0 is asymptotically stable. Rewrite Eq.(3.37)as

2 2 2 2

4 2 1 2 2 3 4 0

V =v = − e − −e e −e < (3.38)

which is a negative semi-definite function of e1, e2, e , 3 e4, a, b, c, d, e, and

f ,while from Eqs(3.14),(3.21),(3.27),(3.34)

2 2 2 2 1 1 1 1 4 2 1 2 1 2 2 3 2 3 4 2 2 2 2 2 2 1 2 ( ) ( ) ( ) V v e e e e e e a b c d e f = = + + + + + + + + + + + (3.39) is a positive definite function of e₁, e₂, e , ₃ e₄, a, b, c, d, e, and f .

The Lyapunov asymptotical stability theorem can not be satisfied in this case. The common origin of error dynamics and parameter update dynamics cannot be determined to be asymptotically stable. By pragmatical asymptotical theorem (see Appendix ) D is a 10 -manifold, n = 10 and the number of error state variables p = 4.

Whene₁ = e₂ = e = ₃ e₄= 0 and

a

, b, c, d, e, f take arbitrary values , V = , 0

so X is a 6-dimational space, m = n – p = 10 – 4 = 6. m + 1≤ n are satisfied. By the pragmatical asymptotical stability theorem, not only error vector e tends to zero but also all estimated parameters approach their uncertain parameters. PHPHCGS of chaotic systems by adaptive backstepping control is accomplished. The equilibrium point e₁ =

2

e = e = ₃ e₄ = a = b = c = d = e = f = 0 is pragmatically asymptotically

stable. The generalized synchronization is accomplished after 2000s with g = -0.1, ₁ 2

g = 4, g = -0.1, and ₃ g = 0.2 while ₄ e₁(0)= −2.4001, e₂(0)= −85.9999,

3(0) 3.0001,

e = − and e₄(0)= −1.9999. The estimated parameters approach the uncertain

parameters of the chaotic system as shown in Figs 3.2-3.8. The initial values of estimated parameters areaˆ(0)=bˆ(0)=cˆ(0)=dˆ(0)=eˆ(0)= 0.

Fig. 3.1 Chaotic phase portrait of new Duffing – Van der Pol system.

Fig. 3.3 The time histories of estimated parameter ˆa .

Fig. 3.5 The time histories of estimated parameter ˆc .

Fig. 3.7 The time histories of estimated parameter ˆe.

Chapter 4

Pragmatical Hybrid Projective and Symplectic

Synchronization of Different Order Systems with New

Control Lyapunov Function by Adaptive Backstepping

Control

4.1Preliminaries

In this Chapter, a new kind of synchronization and a new control Lyapunov function for backstepping are proposed. The symplectic* synchronization

( , , , )

y=H x y z t (4.1)

is studied, where x and y are the state vectors of the ‘‘master’’ and of the ‘‘slave’’ respectively, and z is a given function vector of time, which may take various forms, either a regular or a chaotic function of time [28]. When

( , , , ) ( , , )

y=H x y z t =H x z t (4.2)

it reduces to the generalized synchronization. Therefore symplectic [72] symchronization is an extension of generalized synchronization.

In Eq. (4.1), the final desired state y of the ‘‘slave’’ system not only depends upon the“master’’ system state x, but also depends upon the ‘‘slave’’ system state y itself. Therefore the ‘‘slave’’ system is not a traditional pure slave completely obeying the

*The term ‘‘symplectic’’ comes from the Greek for ‘‘interwined’’. H. Weyl first introduced the term in 1939 in his book “The Classical Groups＂(P. 165 in both the first edition, 1939, and second edition, 1946, Princeton University Press)

‘‘master’’ system but plays a role to determine the final desired state of the ‘‘slave’’ system. In other words, it plays a ‘‘interwined’’ role, so we call this kind of synchronization ‘‘symplectic synchronization’’, and call the“master’’ system partner A, the ‘‘slave’’ system partner B.

The application of symplectic synchronization has great potential. For instance, if the symplectically synchronized chaotic signal is used as the signal carrier from a transmitter, secure communication is more difficult to be deciphered.

In this Chapter, a new control Lyapunov function ( ) exp( T ) 1

V e = ke e − (4.3)

is proposed for backstepping control where e is error dynamics. Using the new control Lyapunov function, error tolerance can be decreased marvelously as comparing to that obtained by traditional control Lyapnov function

( ) T

V e =e e (4.4)

This Chapter is organized as follows. In Section 2, by the GYC pragmatical asymptotical stability theorem with new control Lyapunov functions, a pragmatical hybrid projective and symplectic synchronization scheme is achieved. In Section 3, chaos in the Mathieu - Duffing system and Lusystem [36] with four dimensions are given. In Section 4, using the new control Lyapunov functions numerical results of PHPSS by adaptive backstepping control is achieved.

4.2. Symplectic synchronization scheme

Assume that there are two different nonlinear chaotic dynamical systems and that the partner A controls the partner B partially. The partner A is given by

( )

x= f x (4.5)

where

(

₁, , ,₂

)

T _n n

x= x x x ∈R denotes the state vector and f a vector function.

The partner B is given by

( )

y=g y (4.6)

where

(

₁, , ,₂

)

T _n n

y= y y y ∈R denotes the state vector, g is another vector-function.

The controlling partner B becomes

y_u =g y( )+u t( ) (4.6b) where ( )

(

₁( ), ( ), , ( )₂

)

T _n

n

u t = u t u t u t ∈R is a control input vector.

( )

z=h z (4.7)

is a functional system, where ( , ,...., )_{1 2} T n n

z= z z z ∈ℜ is a state vector which is either

chaotic or regular function vector of time, h is a given vector function.

Our goal is to design the controller u(t) so that the state vector y of the partner B asymptotically approaches ( , , , )H x y z t and finally to accomplish the synchronization in

the sense that the limit of the error vector ( ) ( , ,_{1 2} , )T n e t = e e e approaches zero: 0 lim = ∞ → e t (4.8) where e= y_u −H x y z t( , , , ) (4.9) From Eq. (4.9) , it is obtained that

e y_u H x H y H z H x y z t

∂ ∂ ∂ ∂

= − − − −

∂ ∂ ∂ ∂ (4.10) By Eq. (4.5), Eq. (4.6), Eq. (4.6b), Eq. (4.7), and Eq. (4.9) becomes

e g y( ) u t( ) H f x( ) H g y( ) H h z( ) H

x y y t

∂ ∂ ∂ ∂

= + − − − −

A new control Lyapnuov function V(e) is chosen as a positive definite function of e:

( ) exp( T ) 1

V e = ke e − (4.12)

where k is a positive constant. Its time derivative along any solution of Eq.(4.11)becomes ( ) 2 T{ ( ) ( ) H ( ) H ( ) H ( ) H}exp( T ) V e ke g y u t f x g y h z ke e x y z t ∂ ∂ ∂ ∂ = + − − − − ∂ ∂ ∂ ∂ (4.13)

When )u(t is chosen so that 2 T exp( T ) n n

V = ke C _× e ke e where C_{n n×} is a diagonal

negative definite constant matrix, then V is a negative definite function of e. By

Lyapunov theorem of asymptotical stability 0

lim = ∞ → e

t (4.14) The symplectic synchronization is obtained [22-24].

4.3. Lü system

Lusystem can be described as follows [36]:

1 1 2 1 2 1 3 1 1 3 1 2 1 3 ( ) d z a z z dt d z z z c z dt d z z z b z dt ⎧ _{= −} ⎪ ⎪ ⎪ _{= − +} ⎨ ⎪ ⎪ _{= −} ⎪⎩ (4.15)

where z ,₁ z ,₂ z are state variables, and ₃ a b c are parameters. When the parameter ₁, ,_{1 1}

1 36, 1 3,

a = b = and c₁=15, the system dynamics is chaotic. We choose Lu system of

three states as the different order system for four states new Mathieu – Duffing system. The additional fourth equation is chosen as

z₄ = + + (4.16) z₁ z₂ z₃

d z₄ a z( ₂ z₁) z z_{1 3} cz₁ z z_{1 2} bz₃

dt = − − + + − (4.17)

Augmented Lusystem with four states becomes:

1 1 2 1 2 1 3 1 1 3 1 2 1 3 4 1 2 1 1 3 1 1 1 2 1 3 ( ) ( ) d z a z z dt d z z z c z dt d z z z b z dt d z a z z z z c z z z b z dt ⎧ _{= −} ⎪ ⎪ ⎪ _{= − +} ⎪ ⎨ ⎪ _{= −} ⎪ ⎪ ⎪ = − − + + − ⎩ (4.18)

where z ,₁ z ,₂ z ,₃ z are state variables, and ₄ a b c are parameters. When ₁, ,_{1 1}

1 36, 1 3,

a = b = and c₁=15, the system dynamics is chaotic as shown in Fig.4.1

4.4. Numerical results of CHPS by adaptive backstepping control

Take Mathieu-Duffing system (2.3) as master system, where a, b, c, d, ,

e f are uncertain parameters and following Mathieu-Duffing system as slave system:

1 2 3 2 3 1 3 1 2 3 3 4 3 4 3 3 4 1 ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ d y y dt d y a by y a by y cy dy dt d y y dt d y y y ey fy dt ⎧ ₌ ⎪ ⎪ ⎪ _{= − + − + − +} ⎪ ⎨ ⎪ ₌ ⎪ ⎪ ⎪ = − − − + ⎩ (4.19)

where a b c d e and fˆ, , , ,ˆ ˆ ˆ ˆ ˆare estimated parameters. Finally, take Eq.(4.19) as functional system.

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 ( , , , ) ( , , , ) ( , , , ) ( , , , ) H x y z t g x z y H x y z t g x z y H x y z t g x z y H x y z t g x z y = ⎧ ⎪ ₌ ⎪ ⎨ ₌ ⎪ ⎪ ₌ ⎩ (4.20) Eq.(4.9) becomes 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 e y g x z y e y g x z y e y g x z y e y g x z y = − ⎧ ⎪ = − ⎪ ⎨ = − ⎪ ⎪ = − ⎩ (4.21)

where g g g and g are partly positive and partly negative constants in order to ₁, , ,_{2 3 4} form hybrid projective synchronization.

In order to lead( , , , )y y y y_{1 2 3 4} to(g x z y g x z y g x z y g x z y , _{1 1 1 1}, _{2 2 2 1}, _{3 3 3 3}, _{4 4 4 4}) u u u ₁, ,_{2 3}

and u₄ are added to Eq.(4.19) to form the partner B in Eq.(4.6b), respectively:

1 2 1 3 2 3 1 3 1 2 3 2 3 4 3 3 4 3 3 4 1 4 ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ d y y u dt d y a by y a by y cy dy u dt d y y u dt d y y y ey fy u dt ⎧ _{= +} ⎪ ⎪ ⎪ _{= − + − + − + +} ⎪ ⎨ ⎪ _{= +} ⎪ ⎪ ⎪ = − − − + + ⎩ (4.22) By Eq.(4.10) i ui i i i i i i i i i i i i e = y −g x z y −g x z y −g x z y ,(i=1, 2,3, 4) (4.23)

1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 3 3 2 1 1 2 3 3 3 3 3 3 3 3 1 3 3 3 3 1 3 3 2 2 2 2 1 1 3 2 1 2 2 2 1 3 2 2 2 1 2 2 3 2 1 3 2 2 2 3 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) e e g x z y g x z y g a x z y g a x z y g x z y u e ay ay cy dg x z y bg x z y y bg x z y y cg x z y d by by e ag x z y bg x x z y ag x z y bg x x z y dg x z y = + − − + − + = − − − + − − + + − − + + + + − _{2 2 1 3 2 1 2 2 2 2 2 2 2 1} 3 3 2 2 2 1 3 2 2 2 1 2 2 2 1 3 2 2 2 2 2 2 2 3 2 3 4 4 4 4 4 3 4 3 3 3 3 1 2 3 3 1 3 3 3 3 1 3 3 4 3 3 3 4 3 3 4 1 4 3 4 4 4 3 4 4 ˆ ˆ _ˆ ˆ _ˆ ˆ ˆ ˆ g x z z y b g x z y ag x z y bg x z y y ag x z y bg x z y y cg x z y dg x z y u e e g x z y g x z y g x z z y g c x z y g c x z y u e y y ey fy g x z y g x z y eg + − + + + + + − + = + − − + − + = − − − + + + + _{4 4 4 4 4 1 4 4} 1 4 4 2 4 1 4 4 1 4 4 4 1 3 4 1 4 4 2 4 4 4 1 2 4 3 1 4 4 3 4 4 4 4 3 ˆ 4 4 4 4 ˆ 4 4 4 1 4 x z y fg x z y a g x z y a g x z y g x z z y b g x z y g x z z y c g x z y g x z y eg x z y fg x z y u ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ₋ ⎪ − + + − − ⎪ ⎪ + + + − + ⎪⎩ (4.24)

Step1. We consider the asymptotical stability of e₁=0:

1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1

e = +e g x z y −g x z y −g a x z y +g a x z y −g x z y + (4.25) u

where

e

₂ is regarded as a controller. Choose a control Lyapunov function of the form 2

1 exp( 1 1) 1 0

v = k e − > (4.26)

Its time derivative along the solution of Eq.(4.25) is

1 1 1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 2 1 2 1 2 1 1 1 2 ( ) exp( ) v k e e g x z y g x z y g a x z y g a x z y g x z y u k e = + − − + − + (4.27) Assume virtual controllere₂ =α₁( )e₁ = −e₁. Eq.(4.27) can be rewritten as

1 1 1 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 2 1 2 1 2 1 1 1 2 ( ) exp( ) v k e e g x z y g x z y g a x z y g a x z y g x z y u k e = − + − − + − + (4.28) Choose 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 u = −g x z y +g x z y +g a x z y −g a x z y +g x z y (4.29) Eq.(4.27) becomes 2 2 1 2 1 1 exp( 1 1) 0 v = − k e k e < (4.30)

Step2. When

e

₂ is considered as a controller,

α

₁

(

e

₁

)

is an estimative function. Define

2 2 1( )1 1 2

w = −e α e = +e e (4.31)

Study the ( ,e w_{2 2}) system. We have

2 1 2

w = +e e (4.32)

Choose a control Lyapunov function of the form 2 1 2 2 2 2

2 1 exp( 2 2) 2( ) 1 0

v = +v k w + a +b +c +d − > , (4.33)

where a= −a aˆ,b= −b bˆ,c= −c cˆ,d = −d dˆ and ˆa,ˆb, ˆc, ˆd are estimated values of

the unknown parameters a, b, c, d, respectively. We have 2 2 1 2 2 1 2 2 2 3 3 1 2 2 1 1 2 3 3 3 3 3 3 3 3 1 3 3 3 3 1 3 3 2 2 2 2 2 1 2 2 2 1 3 2 2 2 1 2 2 2 1 3 2 2 2 3 2 2 2 2 1 3 2 2 ( ) exp( ) ˆ ˆ ˆ ˆ ˆ ˆ 2 ( v v k w e e k w aa bb cc dd v k w ay ay cy dg x z y bg x z y y bg x z y y cg x z y ag x z y bg x x z y ag x z y bg x x z y dg x z y g x z z y b = + + + + + + = + − − − + − − + + + + + − + − _{1 2 2 2 2 2 2 2 1 2 2 2 1 3} 3 3 2 2 2 2 1 2 2 2 1 3 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2 2 3 1 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ) exp( ) ˆ ˆ ˆ 2 ( ) exp( ) g x z y ag x z y bg x z y y ag x z y bg x z y y cg x z y dg x z y e u k w aa bb cc dd k w e d by by k w + + + + + − + + + + + + + − − (4.34)

Choose controllere₃ =α₂( ) 0e₁ = , and choose

2 3 2 2 2 2 2 1 2 2 2 1 2 2 2 3 2 2 2 2 2 1 3 2 2 2 3 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 3 2 1 1 2 3 3 ˆ 2 exp( )( ) ˆ 2 exp( )( ) ˆ 2 exp( )( ) ˆ 2 exp( )( ) ˆ ˆ ˆ ˆ a a k w k w g x z y g x z y a b b k w k w g x x z y g x x z y b c c k w k w g x z y c d d k w k w g x z y d u ay ay cy dg x = − = − − + = − = − − + = − = − + = − = + = + + − 3 3 3 3 3 3 3 1 3 3 3 3 1 3 3 2 2 2 2 2 1 2 2 2 1 3 2 2 2 1 2 2 2 1 3 2 2 2 3 2 2 2 2 1 3 2 1 2 2 2 2 2 2 2 1 2 2 2 1 3 3 3 2 2 2 1 2 2 2 1 3 2 2 2 2 2 2 2 3 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ _ˆ ˆ ˆ ˆ ˆ ˆ z y bg x z y y bg x z y y cg x z y ag x z y bg x x z y ag x z y bg x x z y dg x z y g x z z y b g x z y ag x z y bg x z y y ag x z y bg x z y y cg x z y dg x z y e + + − − − − + + − + − − − − − + − 2 2 2 2 2 a b c d w ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + + + + − ⎪⎩ (4.35) Eq.(4.33) becomes

2 2 2 1 2 2 2 exp( 2 2) 0

v = −v k w k w < (4.36)

This means that e₂ =0 is asymptotically stable.

Step3. When e is considered as a controller, ₃ α₂( )e₁ is estimative function.

Define

3 3 2( )1 3

w = −e α e = (4.37) e

Its time derivative is 3 3

w = (4.38) e

Choose a control Lyapunov function of the form 2

3 2 exp( 3 3) 1 0

v = +v k w − > (4.39)

Its time derivative through the third equation of Eq.(4.39) is

3 2 3 3 4 4 4 4 4 3 4 3 3 3 3 1 2 3 3 3 3 3 3 2 3 3 3 3 4 3 3 3 2 ( ) exp( ) v v k w e g x z y g x z y g x z z y g a x z y g a x z y u k w = + + − − + − + (4.40)

where e₄ is a virtual controller. Take

4 3( )1 3 3 e =α e = −w = − (4.41) e and choose 3 4 4 4 4 3 4 3 3 3 3 1 2 3 3 1 3 3 3 3 1 3 3 4 u = −g x z y +g x z y +g x z z y −g c x z y +g c x z y (4.42) Eq.(4.40) becomes 2 2 3 3 2 3 3 exp( 3 3) 0 v = −v k w k w ≤ (4.43)

This means that e₃= is asymptotically stable. 0

Step4. When e₄ is a virtual controller, α₃( )e₁ is estimative function.

Define 4 3 3( )1 3 4 w = −e α e = + (4.44) e e then 4 3 4 w = + (4.45) e e

Choose a control Lyapunov function of the form 2 1 2 2

4 3 exp( 4 4) 2( ) 1 0

V =v = +v k w + e + f − > (4.46)

Its time derivative is

3 3 4 3 4 4 4 3 3 4 1 4 3 4 4 4 3 4 4 4 4 4 4 4 1 4 4 1 4 4 2 4 1 4 4 1 4 4 4 1 3 4 3 1 4 4 2 4 4 4 1 2 4 1 4 4 3 4 4 4 4 3 4 4 4 4 2 4 4 4 1 4 4 4 ˆ ˆ 2 ( ˆ ˆ _{) exp( )} V v v k w e y y ey fy g x z y g x z y eg x z y fg x z y a g x z y a g x z y g x z z y b g x z y g x z z y c g x z y g x z y eg x z y fg x z y u k w ee ff = = + − − − + + + + − − + + − − + + + − + + + (4.47) Choose 2 4 4 4 4 4 4 4 4 2 4 4 4 4 4 1 4 4 3 3 4 4 3 3 4 1 4 3 4 4 4 3 4 4 4 4 4 4 4 1 4 4 1 4 4 2 4 1 4 4 1 4 4 4 1 3 4 1 4 4 2 4 4 4 1 2 4 1 4 4 ˆ 2 exp( )( ) ˆ 2 exp( )( ) ˆ ˆ e e k w k w g x z w e f f k w k w g x z w f u e y y ey fy g x z y g x z y eg x z y fg x z y a g x z y a g x z y g x z z y b g x z y g x z z y c g x z = − = − + = − = + = − + + + − − − − + + − − + + − 3 3 4 4 4 4 3 4 4 4 4 2 2 4 4 4 1 4 ˆ ˆ y g x z y eg x z y fg x z y e f w ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ _{− −} ⎪ ⎪ _{+ + + −} ⎩ (4.48) Eq.(4.47) becomes 2 2 4 3 4 4 4 4 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 2 exp( ) 0 V v v k w k w k e k e k w k w k w k w k w k w = = − = − − − − < (4.49)

This means that e₄ =0 is asymptotically stable. Rewrite Eq.(4.49) as

2 2 2 2 4 1 1 1 1 2 1 2 2 1 2 2 2 2 2 3 3 3 3 4 3 4 4 3 4 2 exp( ) 2 ( ) exp( ( ) ) 2 exp( ) 2 ( ) exp( ( ) ) 0 V v k e k e k e e k e e k e k e k e e k e e = = − − + + − − + + < (4.50)

which is a negative semi-definite function ofe₁, e₂, e , ₃ e₄, a, b, c, d, e, and

f ,while from Eqs(4.26),(4.33),(4.39),(4.46).

2 2 2 2

4 1 1 2 1 2 3 3 4 3 4

2 2 2 2 2 2

1 2

exp( ) exp( ( ) ) exp( ) exp( ( ) )

( ) 4

V v k e k e e k e k e e a b c d e f

= = + + + + +

+ + + + + + − (4.51)

is a positive definite function of e₁, e₂, e , ₃ e₄, a, b, c, d, e, and f .

common origin of error dynamics and parameter update dynamics cannot be determined to be asymptotically stable. By GYC pragmatical asymptotical stability theorem (see Appendix ) D is a 10 -manifold, n = 10 and the number of error state variables p = 4.

Whene₁ = e₂ = e = ₃ e₄= 0 and a, b, c, d, e, f take arbitrary values , V = , 0

so X is of 6 dimensions, m = n – p = 10 – 4 = 6. m + 1≤ n are satisfied. By the GYC pragmatical asymptotical stability theorem, not only error vector e tends to zero but also all estimated parameters approach their uncertain parameters. PHPSS of chaotic systems by adaptive backstepping control is accomplished. The equilibrium point e₁ = e₂ = e ₃

= e₄ = a = b = c = d = e = f = 0 is asymptotically stable.

The numerical results are shown in Figs 4.2-4.6. the numerical data used are: 1

g = 0.0001 , g = 0.0001₂ − , g = -0.0001 , ₃ g = -0.0001 ,₄ k₁=0.01, k₂ =0.01,

3 0.01,

k = and k₄ =0.01, e₁(0)= −2.00014 , (0) 9.99999e₂ = , e₃(0)= −2.00014 and

4(0) 9.99999

e = . Using the new control Lyapunov functions, the error tolerances are

reduced marvelously to ₁₀−17 _{of that using traditional control Lyapnov function} ( ) T

V e =e e as shown in Figs.4.2-4.3. The estimated parameters approach the uncertain

parameters of the chaotic system as shown in Figs. 4.4-4.6. The initial values of estimated parameters are ˆ(0)a = ˆ(0)b = ˆ(0)c = ˆ(0)d = ˆ(0)e = ˆ(0) 0f = . This property makes the

Fig. 4.1 Chaotic phase protract of Lu system.

Fig. 4.2 Synchronization error for traditional control Lyapunov function ( ) T

V e =e e

Fig. 4.3 Synchronization error for new control Lyapunov function ( ) exp( T ) 1

V e = ke e −

from 600s ~ 1000s.

Fig. 4.5 The time histories of estimated parameter c and d.

Chapter 5

Chaos Control of a New Mathieu- Duffing System by

GYC Partial Region Stability Theory

5.1 Preliminaries

In this Chapter, a new strategy to achieve chaos control by GYC partial region stability [33, 34] is proposed. By using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states and the controllers are simpler than traditional controllers and so reduce the simulation error because they are in lower degree than that of traditional controllers.

This Chapter is organized as follows. In Section 2, chaos control scheme by GYC partial region stability theory is proposed. In Section 3, numerical Simulations of chaos control of new Mathieu – Duffing systems as simulated examples by GYC are achieved.

5.2 Chaos control scheme

Consider the following chaotic systems ( , )t = x f x (5.1) where

[

₁, , ,₂

]

T _n n x x x R = ∈

x is a the state vector, :f R₊×Rn →Rn is a vector

function.

( , )t = y g y (5.2) where

[

₁, , ,₂

]

T _n n y y y R = ∈

y is a state vector, :g R₊×Rn →Rn is a vector function.

In order to make the chaos state x approaching the goal state y , define = −e x y

as the state error. The chaos control is accomplished in the sense that: lim lim( ) 0

t→∞e=t→∞ x y− = (5.3)

In this Chapter, we will use examples in which the e state is placed in the first quadrant of coordinate system and use on partial region stability theory. The Lyapunov function is a simple linear homogeneous function of states and the controllers are simpler because they are in lower degree than that of traditional controllers and so reduce the simulation error because they are in lower degree than that of traditional controllers.

5.3 Numerical simulations of chaos control by GYC

The following chaotic system is a Mathieu – Duffing system of which the old origin is translated to ( , , , ) (50, 50, 50, 50)x x x x_{1 2 3 4} = : 1 2 3 2 3 1 3 1 2 3 3 4 3 4 3 3 4 1 50 ( ( 50))( 50) ( ( 50))( 50) ( 50) ( 50) 50 ( 50) ( 50) ( 50) ( 50) d x x dt d x a b x x a b x x dt c x d x d x x dt d x x x e x f x dt ⎧ _{= −} ⎪ ⎪ ⎪ _{= − + − − − + − −} ⎪ ⎪ _{− − + −} ⎨ ⎪ ⎪ = − ⎪ ⎪ = − − − − − − + − ⎪ ⎩ (5.4)

and the chaotic motion always happens in the first quadrant of coordinate system 1 2 3 4

examples where the initial conditions is x₁(0) 49,= x₂(0) 61,= x₃(0) 49,=

4(0) 61

x = . The chaotic motion is shown in Fig. 5.1.

In order to lead ( , , , )x x x x_{1 2 3 4} to the goal, we add control terms u1, u2 and u3 to

each equation of Eq. (5.4), respectively.

1 2 1 3 2 3 1 3 1 2 3 2 3 4 3 3 4 3 3 4 1 4 50 ( ( 50))( 50) ( ( 50))( 50) ( 50) ( 50) 50 ( 50) ( 50) ( 50) ( 50) d x x u dt d x a b x x a b x x dt c x d x u d x x u dt d x x x e x f x u dt ⎧ _{= − +} ⎪ ⎪ ⎪ _{= − + − − − + − −} ⎪ ⎪ _{− − + − +} ⎨ ⎪ ⎪ = − + ⎪ ⎪ = − − − − − − + − + ⎪ ⎩ (5.5)

CASE I. Control the chaotic motion to zero.

In this case we will control the chaotic motion of the Mathieu – Duffing system (5.4) to zero. The goal is y= . The state error is e x y x0 = − = and error dynamics becomes

2 2 1 1 2 1 1 1 3 2 2 3 1 3 1 2 2 2 3 1 1 2 2 2 3 3 4 3 3 3 3 2 2 4 4 3 3 4 1 3 3 4 50 ( ( 50))( 50) ( ( 50))( 50) ( 50) ( 50) 50 ( 50) ( 50) ( 50) ( 50) e x x e e u e x a b x x a b x x c x d x e e u e x x e e u e x x x e x f x e e u = = − + − + = = − + − − − + − − − − + − + − + = = − + − + = = − − − − − − + − + − + (5.6)

In Fig. 5.1, we see that the error dynamics always exists in first quadrant.

By GYC partial region stability, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

V = + + + (5.7) e e e e

1 2 3 4 2 2 3 2 1 1 1 3 1 3 1 2 2 2 2 2 3 1 1 2 4 3 3 3 3 2 2 3 3 4 1 3 3 4 ( 50 ) ( ( ( 50))( 50) ( ( 50))( 50) ( 50) ( 50) ) ( 50 ) ( ( 50) ( 50) ( 50) ( 50) ) V e e e e x e e u a b x x a b x x c x d x e e u x e e u x x e x f x e e u = + + + = − + − + + − + − − − + − − − − + − + − + + − + − + + − − − − − − + − + − + (5.8) Choose 2 1 2 1 1 3 2 3 1 3 1 2 2 3 1 2 2 3 4 3 3 3 2 4 3 3 4 1 3 4 ( 50) ( ( ( 50))( 50) ( ( 50))( 50) ( 50) ( 50)) ( 50) ( ( 50) ( 50) ( 50) ( 50)) u x e e u a b x x a b x x c x d x e e u x e e u x x e x f x e e = − − + − = − − + − − − + − − − − + − − − = − − + − = − − − − − − − + − − − (5.9) We obtain 1 2 3 4 0 V = − − − − < (5.10) e e e e

which is negative definite function. The numerical results are shown in Fig.5.2. After 30 sec, the motion trajectories approach the origin.

CASE II. Control the chaotic motion to a sine function.

In this case we will control the chaotic motion of the Mathieu – Duffing system (5.5) to sine function of time. The goal is y=msinwt. The error equation

sin

e= − = −x y x m wt (5.11)

lim _i lim( _{i i}sin _i ) 0, 1, 2,3, 4

t→∞e =t→∞ x −m w t = i= (5.12) and cose_i = −x_i w m_{i i} w t_i (i=1, 2,3, 4), and m₁ =m₂ =m₃ =m₄. The error dynamics is